Mathematical Sciences: Stochastic Models for Reliability of Systems with Dependencies Among Components

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Proposal: DMS 9503104 PIs: W. J. Padgett, J. D. Lynch and S. D. Durham Institution: University of South Carolina - Columbia Title: STOCHASTIC MODELS FOR RELIABILITY OF SYSTEMS WITH DEPENDENCIES AMONG COMPONENTS Abstract: The research involves the reliability of complex systems. There are three main thrusts: (1) modeling component reliability, (2) incorporating component reliability and dependencies into the system reliability which result in tractable data analytic models, and (3) developing a criticality theory for such models. Regarding (1), some models are studied for the analysis of component reliability, including the conditional Weibull and inverse Gaussian distributions and a Poisson-Weibull flaw model. The "conditional Weibull" distribution is studied since it fits certain fiber strength data sets well and may be justified due to "censoring" considerations in the fiber manufacturing and testing processes. The Poisson-Weibull flaw model with finite-state Markov random intensity has the mixed distribution (zero intensity) and the mixed hazard model (infinite intensity) as extremes. A major objective of this part of the project is the investigation of a unified mixture theory for this flaw model which subsumes the theory of mixtures developed for the extremes of mixed distributions and mixed hazards, respectively. To address (2) and (3), a general model is investigated which is hierarchical in nature. The hierarchy consists of (i) the micro (or component) level, (ii) the subsystem (or "bundle of components") level, and (iii) the system (or "chain of bundles") level. It has long been known that in many systems, the failure of a component changes the stress applied to the remaining components. Thus, incorporating component dependencies into a reliability model in a realistic manner is highly desirable for accurate results. Here, component dependencies/interactions are incorporated into the model by using "load-sharing rules," m ost applicable to situations where "loadings" are due to mechanical or physical considerations. General monotone load-sharing rules are considered for which a method of calculating the system reliability has been developed. The present research includes a special class of these rules where the component load can be calculated using absorption probabilities for random walks on a network. In particular, a number of the popular load-sharing rules can be reduced to the consideration of electrical networks. Energy considerations give insight into the behavior of stress concentrations induced by these rules as components fail. Major objectives are to continue the investigation of these electrical network rules, and other network rules which are appropriate for mechanical load transfer situations, especially for composite materials, and, specifically to study the "effective distance" that a load can be transferred via matrix material and its relationship to the "ineffective length" around fiber breaks for load transfer through the matrix. Criticality issues are also considered. Preliminary work for k-out-of-n systems and systems with a small number of components suggest that an exact theory may be obtainable when each component has a Weibull failure distribution using a mixed transformed gamma model, where the transformation depends on the Weibull distribution. Such a model also lends itself to calculation of extreme value approximation errors and to system identification using the mixing distribution. The research involves the development of models for describing the failure of complex systems with dependent components. The researchers are investigating reliability models for complex systems of components including complex materials which allow for component dependencies and interactions. Such models have direct application to many problems of current importance, including failure of fibrous composite materials and electrical networks. This has important imp lications for the design and large scale manufacture of complex materials where high reliability.